Report

Boltzmann Machines and their Extensions S. M. Ali Eslami Nicolas Heess John Winn March 2013 Heriott-Watt University Goal Define a probabilistic distribution on images like this: 2 What can one do with an ideal shape model? Segmentation 3 Weizmann horse dataset Sample training images 327 images 4 What can one do with an ideal shape model? Image 5 What can one do with an ideal shape model? Computer graphics 6 Energy based models Gibbs distribution 7 Shallow architectures Mean 8 Shallow architectures MRF 9 Existing shape models Most commonly used architectures Mean MRF sample from the model sample from the model 10 What is a strong model of shape? We define a strong model of object shape as one which meets two requirements: Realism Generalization Generates samples that look realistic Can generate samples that differ from training images Training images Real distribution Learned distribution 11 Shallow architectures HOP-MRF 12 Shallow architectures RBM 13 Shallow architectures Restricted Boltzmann Machines The effect of the latent variables can be appreciated by considering the marginal distribution over the visible units: 14 Shallow architectures Restricted Boltzmann Machines In fact, the hidden units can be summed out analytically. The energy of this marginal distribution is given by: where 15 Shallow architectures Restricted Boltzmann Machines All hidden units are conditionally independent given the visible units and vice versa. 16 RBM inference Block-Gibbs MCMC 17 RBM inference Block-Gibbs MCMC 18 RBM learning Stochastic gradient descent Maximize with respect to 19 RBM learning Contrastive divergence Getting an unbiased sample of the second term, however is very difficult. It can be done by starting at any random state of the visible units and performing Gibbs sampling for a very long time. Instead: 20 RBM inference Block-Gibbs MCMC 21 RBM inference Block-Gibbs MCMC 22 RBM learning Contrastive divergence • Crudely approximating the gradient of the log probability of the training data. • More closely approximating the gradient of another objective function called the Contrastive Divergence, but it ignores one tricky term in this objective function so it is not even following that gradient. • Sutskever and Tieleman have shown that it is not following the gradient of any function. • Nevertheless, it works well enough to achieve success in many significant applications. 23 Deep architectures DBM 24 Deep architectures Deep Boltzmann Machines 25 Deep architectures Deep Boltzmann Machines Conditional distributions remain factorised due to layering. 26 Shallow and Deep architectures Modeling high-order and long-range interactions MRF RBM DBM 27 Deep Boltzmann Machines DBM • Probabilistic • Generative • Powerful Typically trained with many examples. We only have datasets with few training examples. 28 From the DBM to the ShapeBM Restricted connectivity and sharing of weights DBM ShapeBM Limited training data, therefore reduce the number of parameters: 1. 2. 3. Restrict connectivity, Tie parameters, Restrict capacity. 29 Shape Boltzmann Machine Architecture in 2D Top hidden units capture object pose Given the top units, middle hidden units capture local (part) variability Overlap helps prevent discontinuities at patch boundaries 30 ShapeBM inference Block-Gibbs MCMC image reconstruction sample 1 sample n Fast: ~500 samples per second 31 ShapeBM learning Stochastic gradient descent Maximize with respect to 1. Pre-training • Greedy, layer-by-layer, bottom-up, • ‘Persistent CD’ MCMC approximation to the gradients. 2. Joint training • Variational + persistent chain approximations to the gradients, • Separates learning of local and global shape properties. ~2-6 hours on the small datasets that we consider 32 Results Sampled shapes Evaluating the Realism criterion FA Incorrect generalization RBM Failure to learn variability ShapeBM Data Weizmann horses – 327 images – 2000+100 hidden units Natural shapes Variety of poses Sharply defined details Correct number of legs (!) 34 Sampled shapes Evaluating the Realism criterion Weizmann horses – 327 images – 2000+100 hidden units This is great, but has it just overfit? 35 Sampled shapes Evaluating the Generalization criterion Weizmann horses – 327 images – 2000+100 hidden units Sample from the ShapeBM Closest image in training dataset Difference between the two images 36 Interactive GUI Evaluating Realism and Generalization Weizmann horses – 327 images – 2000+100 hidden units 37 Further results Sampling and completion Caltech motorbikes – 798 images – 1200+50 hidden units Training images ShapeBM samples Sample generalization Shape completion 38 Constrained shape completion Evaluating Realism and Generalization ShapeBM NN Weizmann horses – 327 images – 2000+100 hidden units 39 Further results Constrained completion ShapeBM NN Caltech motorbikes – 798 images – 1200+50 hidden units 40 Imputation scores Quantitative comparison Weizmann horses – 327 images – 2000+100 hidden units 1. Collect 25 unseen horse silhouettes, 2. Divide each into 9 segments, 3. Estimate the conditional log probability of a segment under the model given the rest of the image, 4. Average over images and segments. Score Mean RBM FA ShapeBM -50.72 -47.00 -40.82 -28.85 41 Multiple object categories Simultaneous detection and completion Caltech-101 objects – 531 images – 2000+400 hidden units Train jointly on 4 categories without knowledge of class: Shape completion Sampled shapes 42 What does h2 do? Multiple categories Class label information Accuracy Weizmann horses Pose information Number of training images 43 What does h2 do? 44 What does the overlap do? 45 Summary • Shape models are essential in applications such as segmentation, detection, in-painting and graphics. • The ShapeBM characterizes a strong model of shape: – Samples are realistic, – Samples generalize from training data. • The ShapeBM learns distributions that are qualitatively and quantitatively better than other models for this task. 46 Questions MATLAB GUI available at http://arkitus.com/Ali/